axiomatic sets 33 (regularity 2)

Axiom of regularity is also called axiom of foundation.
\[ \forall x[x\ne \phi\rightarrow \exists y[y\in x\wedge y\cap x=\phi]]  \]

This axiom requires that even infinite sets are bounded
and all collections of things is not necessarily a set.

Example(1); $A=\left\{ x: x\notin x \right\}$  is not a set.

$A$  is a collection of a set which does not contain itself.
(you are able to consider that $A$  is a very huge collection. )

If you accept that $A$  is a set, then a contradiction occurs.

We will give you a question.
Which do you prefer, $A\in A$  or $A\notin A$ ?

This is Russell's paradox.

However, using axiom of regularity we have to recognize $A$  is not a set.
If $A\in A$ is accepted, as there exists the sequence $A\ni A\ni A\ni\cdots$ ,
$A$ violates the axiom.

If $A\notin A$ is accepted, $\left\{ x: x\notin x \right\}$ is not satisfied.


Example(2); The collection $A$ of all sets is not a set.

$A$ is the largest collection in all sets.

However, as $A\in A$ , there is a sequence $A\ni A\ni A\ni\cdots$ .
Therefore, by axiom of regularity $A$ is not a set.

The collections which are not set are called classes. 

axiomatic sets 32 (regularity)

This axiom of regularity was offered by J.Neumann.

\[ \forall x[x\ne \phi\rightarrow \exists y[y\in x\wedge y\cap x=\phi]]  \]

All sets must satisfy this axiom.
A collection of things which can not satisfy this axiom is not a set.

At a glance, you might see it difficult to understand.
This axiom requires that infinite sets are not unbounded.

Suppose a infinite sequence of the sets $a_1\ni a_2\ni a_3\ni\cdots$ .
If $a_i=\left\{ a_{i+1}  \right\}$ ,then
$a_1=\left\{ a_2 \right\}=\left\{ \left\{ a_3 \right\} \right\}=\cdots $ .

(remember $\mathbb{N}$ or axiom on infinity .)

On the other hand, let us consider $A=\left\{ a_1,a_2,a_3,\cdots \right\}$ .
As this $A$ violates the axiom of regularity, $A$  is not a set.

We have to call a class (not a set) $A$ .

Although $\mathbb{N}$ is a set, $\left\{ \infty\right\}$ is not a set.
(please do not misunderstand. )

This axiom makes too big huge collections of things be not a set.

This axiom is equivalent to
" for all $x$ , there does not exist the sequence $x\ni x_1\ni x_2\ni x_3\ni\cdots$ ",
or $x$ such that $x\ni x_1\ni x_2\ni x_3\ni\cdots$  is not a set.